RUI: Pure and Applied Knot Theory: Skeins, Hyperbolic Volumes, and Biopolymers

纽结理论是对连续变形的线圈缠结的数学研究。一个人可以通过一根缠绕的绳子并连接端点来创建一个结，如果一个人可以连续地变形为另一个，例如通过弯曲，拉伸和将线穿过其他线，但不会以任何方式切断或断开绳子，那么两个结是等效的。 这个项目考虑了数学纽结理论的理论问题和应用。一组问题研究了一系列与物理学量子场论有关的纽结不变量。更具体地说，该项目旨在了解结的量子不变量如何检测结的几何性质以及与之相关的三维空间。这项研究的数学技术在数学物理和理论拓扑量子计算中具有潜在的应用。另一组问题涉及打结蛋白质和其他生物聚合物的研究应用，其中一些已知与各种疾病有关。该项目使用节点理论技术开发一个模型，可用于量化和局部拓扑复杂性与生物物理过程的关系。该模型也可以用于潜在的设计合成生物聚合物具有特殊的生物物理特性。该项目包括一些适合与本科生合作的研究问题，以及寻求更普遍地提高数学兴趣的推广和传播活动。PI过去曾成功地让本科生参与类似的研究，并将继续建议和鼓励学生继续从事数学和相关领域的职业。 这项研究分为三个部分，两个部分试图将量子拓扑与双曲几何联系起来，一个部分将纽结理论应用于分子生物学。一个项目涉及一个版本的体积猜想的基础上理论的考夫曼括号绞代数从量子拓扑和其关系的Teichmuller空间的表面从双曲几何。第二个项目研究的代数和几何性质的推广的考夫曼括号代数是相关的装饰Teichmuller空间的表面穿孔。第三个项目涉及与一位生物学家的合作，研究生物聚合物中分子力紧紧抓住的局部缠结。拟议的结理论模型将给出这样的本地纠缠的描述，允许一个量化和测量在实验中的生物聚合物的局部拓扑复杂性的变化。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Knot theory is the mathematical study of entanglement of loops up to continuous deformation. One can create a knot by taking an entangled string and connecting the endpoints, and two knots are equivalent if one can continuously deform one to the other, for example by bending, stretching, and passing strands inside and through others, but without cutting or breaking the string in any way. This project considers both theoretical problems and applications of mathematical knot theory. One group of problems studies a family of invariants of knots related to quantum field theory from physics. More specifically, the project seeks to understand how the quantum invariant of a knot detects geometric properties of the knot and the 3-dimensional spaces that can be associated with it. The mathematical techniques from this research has potential applications to mathematical physics and theoretical topological quantum computing. Another group of problems concerns applications to the study of knotted proteins and other biopolymers, some of which are known to be associated to various diseases. The project uses knot theory techniques to develop a model that can be used to quantify and to relate local topological complexity with biophysical processes. The model can also be used to potentially design synthetic biopolymers with special biophysical properties. The project includes a number of research problems suitable for collaboration with undergraduate students, as well as outreach and dissemination activities that seek to increase interest in mathematics more generally. The PI has successfully involved undergraduate students in similar research in the past and will continue to advise and encourage students to continue careers in mathematics and related areas. The research is split into three parts, two seek to connect quantum topology with hyperbolic geometry and one applies knot theory to molecular biology. One project concerns a version of the Volume Conjecture based the theory of the Kauffman bracket skein algebra from quantum topology and its relationship to the Teichmuller space of a surface from hyperbolic geometry. A second project studies algebraic and geometric properties of a generalization of the Kauffman bracket algebra which is related to the decorated Teichmuller space of a surface with punctures. A third project involves a collaboration with a biophysicist to study local entanglements that are held tightly in place by molecular forces in biopolymers. The proposed knot-theoretic model would give a description of such local entanglements, allowing one to quantify and measure changes in the local topological complexity of biopolymers in experiments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.